Modular Heights of Unitary Shimura Varieties I: Derivatives of Eisenstein Series

2025/09/29 by Ziqi Guo, Guo, Ziqi
Mathematics · #11G40 11G50 14G10 14G35 14G40 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.24363

openalex publication_date 2025/09/29 · openalex created_date 2025/10/19 · openalex updated_date 2026/08/01

Abstract

This is the first of a series of three papers, in which we prove a formula expressing the modular height of a unitary Shimura variety over a CM number field in terms of the logarithmic derivative of the Hecke L-function associated with the CM extension. The main idea of our proof is to compare the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series and the arithmetic degree of a generating series of divisors on unitary Shimura varieties. In this paper, we compute an explicit expression of the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series.

Related

• Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications
• Modularity of arithmetic special divisors for unitary Shimura varieties (with an appendix by Yujie Xu)
• Arithmetic volumes of unitary Shimura varieties
• Generating series of intersection volumes of special cycles on unitary Shimura varieties
• Modularity of generating series of divisors on unitary Shimura varieties